Remark 1 Assumption REF  is sufficient to uniquely define the controlled process \(X^{\alpha ^1,\alpha ^2}\)  for any control \((\alpha ^1,\alpha ^2)\)  (see below for the definitions). If \(A_1\)  was compact, we would be in the setting of [1]} Chapter IV.4 and these assumptions would guarantee the existence of a smooth value function (in \(\mathcal {C}_b^{1,2} ([0,T]\times \mathbb {R}^d)\) ).

